Coupling a vertex algebra to a large center

Abstract Suppose a Lie group G acts on a vertex algebra $$\mathcal {V}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> . In this article we construct a vertex algebra $${\tilde{V}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>V</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> , which is an extension of $$\mathcal {V}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> by a big central vertex subalgebra identified with the algebra of functionals on the space of regular $$\mathfrak {g}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> -connections $$(\textrm{d}+A)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mtext>d</mml:mtext> <mml:mo>+</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . The category of representations of $${\tilde{\mathcal {V}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>V</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> fibres over the set of connections, and the fibres should be viewed as $$(\textrm{d}+A)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mtext>d</mml:mtext> <mml:mo>+</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -twisted modules of $$\mathcal {V}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> , generalizing the familiar notion of g -twisted modules. In fact, another application of our result is that it proposes an explicit definition of $$(\textrm{d}+A)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mtext>d</mml:mtext> <mml:mo>+</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -twisted modules of $$\mathcal {V}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> in terms of a twisted commutator formula, and we feel that this subject should be pursued further. Vertex algebras with big centers appear in practice as critical level or large level limits of vertex algebras. In particular, we have in mind limits of the generalized quantum Langlands kernel, in which case G is the Langland dual and $$\mathcal {V}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> is conjecturally the Feigin-Tipunin vertex algebra and the extension $${\tilde{\mathcal {V}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>V</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> is conjecturally related to the Kac-DeConcini-Procesi quantum group with big center. With the current article, we can give a uniform and independent construction of these limits.